Various Expressions over Novel k-µ/Inverse Gamma LOS Shadowed Fading

Abstract

This manuscript proposes a novel, very general shadowed k-μ fading model for characterizing the realistic line-of-sight (LOS) propagation scenarios of 5G and beyond. In this model, statistics of the received signal are manifested by the clustering of multipath components. Within each of these clusters, shadowing of the dominant component is represented by the inverse Gamma distribution. First of all, mathematically-tractable expressions of probability density function (PDF) and moment-generating function (MGF) of the k-μ/inverse Gamma LOS-shadowed fading model are obtained. The proposed channel model is validated by reproducing the well-established fading distributions such as Rayleigh, Rice (Nakagami-n) and Nakagami-m under light shadowing conditions and LOS-shadowed fading distributions such as the k-µ/Gamma and Abdi’s Rice/Gamma model under heavy shadowing conditions numerically. The proposed PDF is further utilized to derive the expressions for channel capacity under optimal rate adaptation with constant transmit power (C_ORA), optimal power and rate adaptation (C_OPRA), channel inversion with fixed rate (C_CIFR) and truncated channel inversion (C_CIFR). Furthermore, analytical expressions for outage probability (OP) and error probabilities under different modulation schemes are derived. The analytical expressions are studied under infrequent light, moderate, and frequent heavy shadowing environments used extensively in literature.

Appendices

Appendix A: Showing Steps for Getting Eq. (5) from Eq. (4)

After using Eq. (3) in Eq. (4), we get

$$ f_{{\gamma /{{\uppsi }}}} \left( {\gamma ,{{\uppsi }}} \right) = \frac{{\bar{w}}}{{\bar{\gamma }}}\frac{1}{{2\sigma ^{2} }}\left( {\frac{w}{{{{\uppsi }}d^{2} }}} \right)^{{\frac{{\mu - 1}}{2}}} e^{{ - \frac{{w + {{\uppsi }}d^{2} }}{{2d^{2} }}}} I_{{n - 1}} \left( {\frac{1}{{\sigma ^{2} }}\sqrt {w{{\uppsi }}d^{2} } } \right) $$

(A.1)

Using the relationship $\frac{w}{\overline{w} }=\frac{\gamma }{\overline{\gamma }},$ of the parameters in Eq. (A.1), it can be written as

$$ f_{{\gamma /{{\uppsi }}}} \left( {\gamma ,{{\uppsi }}} \right) = \frac{{\bar{w}}}{{\bar{\gamma }}}\frac{1}{{2\sigma ^{2} }}\left( {\frac{{\bar{w}\gamma }}{{\bar{\gamma }}}\frac{1}{{{{\uppsi }}d^{2} }}} \right)^{{\frac{{\mu - 1}}{2}}} e^{{ - \frac{{\frac{{\bar{w}\gamma }}{{\bar{\gamma }}} + {{\uppsi }}d^{2} }}{{2\sigma ^{2} }}}} I_{{\mu - 1}} \left( {\frac{1}{{\sigma ^{2} }}\sqrt {\frac{{\bar{w}\gamma {{\uppsi }}d^{2} }}{{\bar{\gamma }}}} } \right) $$

(A.2)

Now using the relationship $\overline{w }={d}^{2}+2{\sigma }^{2}\mu$ in Eq. (A.2) and thereafter using $k=\frac{{d}^{2}}{2{\sigma }^{2}\mu }$ we get Eq. (A.3)

after a few mathematical manipulations

$$ f_{{\gamma /{{\uppsi }}}} \left( {\gamma ,{{\uppsi }}} \right) = \frac{{\left( \mu \right)^{{\frac{{\mu + 1}}{2}}} \left( {1 + k} \right)^{{\frac{{\mu + 1}}{2}}} }}{{e^{{{{\uppsi }}\mu k}} \bar{\gamma }}}\left( {\frac{1}{{\mu k}}} \right)^{{\frac{{\mu - 1}}{2}}} \left( {\frac{\gamma }{{\bar{\gamma }}}\frac{1}{{{\uppsi }}}} \right)^{{\frac{{\mu - 1}}{2}}} e^{{ - \mu \left( {1 + k} \right)\frac{\gamma }{{\bar{\gamma }}}}} I_{{\mu - 1}} \left( {\frac{{k2\sigma ^{2} \mu }}{{\sigma ^{2} }}\sqrt {\frac{{\left( {k + 1} \right){{\uppsi }}}}{k}\frac{\gamma }{{\bar{\gamma }}}} } \right) $$

(A.3)

After simple algebraic manipulations, we get Eq. (5)

Appendix B: PDF of k-µ/Inverse Gamma (LOS) Shadowed Fading Model

The unconditional PDF of the k-µ/inverse Gamma (LOS) shadowed fading model is obtained by averaging the conditional PDF in Eq. (5) with PDF of $\uppsi$

$$ f_{\gamma } \left( \gamma \right) = \mathop \smallint \limits_{0}^{\infty } f_{{\gamma /{{\uppsi }}}} \left( {\gamma ,{{\uppsi }}} \right)f_{{{\uppsi }}} \left( {{\uppsi }} \right)d{{\uppsi }} $$

(B.1)

The PDF of the inverse Gamma random variable $\uppsi ,$[18, Eq. (5)] is:

$$ f_{{\uppsi }} \left( {\uppsi } \right) = \frac{{\beta^{\alpha } }}{\Gamma \alpha }\frac{1}{{{\uppsi }^{\alpha + 1} }}e^{{ - \frac{\beta }{{\uppsi }}}} $$

(B.2)

where the shape parameter α and scale parameter β are positive quantities.

Substituting values from Eq. (5) to Eq. (B.2) and thereafter arranging the terms in Eq. (B.1), we get Eq. (B.3)

$$ f_{\gamma } \left( \gamma \right) = \frac{1}{2}\frac{{\mu \left( {1 + k} \right)^{{\frac{\mu + 1}{2}}} }}{{\overline{\gamma }k^{{\frac{\mu - 1}{2}}} }}\left( {\frac{\gamma }{{\overline{\gamma }}}} \right)^{{\frac{\mu - 1}{2}}} e^{{\frac{{ - \mu \left( {1 + k} \right)\gamma }}{{\overline{\gamma }}}}} \frac{{2\beta^{\alpha } }}{\Gamma \alpha }\mathop \smallint \limits_{0}^{\infty } {\uppsi }^{{ - \left( {\alpha + \frac{\mu }{2} + \frac{1}{2}} \right)}} e^{{ - {\uppsi }\mu k}} e^{{ - \left( {\frac{\beta }{{\uppsi }}} \right)}} I_{\mu - 1} \left( {2\mu \sqrt {\frac{{k\left( {1 + k} \right){\uppsi }\gamma }}{{\overline{\gamma }}}} } \right)d{\uppsi } $$

(B.3)

Using the identity $I_{v} \left( z \right) = \mathop \sum \limits_{l = 0}^{\infty } \frac{1}{l!\Gamma v + l + 1}\left( \frac{z}{2} \right)^{v + 2l}$ and thereafter arranging the terms in Eq. (B.3), we get Eq. (B.4)

$$ f_{\gamma } \left( \gamma \right) = \mathop \sum \limits_{l = 0}^{\infty } \frac{1}{{l!\Gamma \left( {\mu + l} \right)}}\frac{1}{2}\frac{{2\beta^{\alpha } }}{\Gamma \alpha }\frac{{\mu \left( {1 + k} \right)^{{\frac{\mu + 1}{2}}} }}{{k^{{\frac{\mu - 1}{2}}} \overline{\gamma }}}\left( {\frac{\gamma }{{\overline{\gamma }}}} \right)^{{\frac{\mu - 1}{2}}} e^{{\frac{{ - \mu \left( {1 + k} \right)\gamma }}{{\overline{\gamma }}}}} \left( {\mu \sqrt {\frac{{k\left( {1 + k} \right)\gamma }}{{\overline{\gamma }}}} } \right)^{\mu - 1 + 2l} \mathop \smallint \limits_{0}^{\infty } t^{{ - \alpha - \frac{\mu }{2} - \frac{1}{2} + \frac{\mu }{2} - \frac{1}{2} + l}} e^{ - t\mu k} e^{{ - {\raise0.7ex\hbox{$\beta $} \!\mathord{\left/ {\vphantom {\beta t}}\right.\kern-0pt} \!\lower0.7ex\hbox{$t$}}}} dt $$

(B.4)

Now, applying the Meijer-G identity corresponding to exponential function ${e}^{-x}={G}^{\begin{array}{cc}1& 0\\ 0& 1\end{array}}\left(x|\begin{array}{c}-\\ 0\end{array}\right)$ in Eq. (B.4), we get Eq. (B.5) after simple algebraic manipulations,

$$ f_{\gamma } \left( \gamma \right) = \mathop \sum \limits_{l = 0}^{\infty } \frac{1}{l!\Gamma \mu + l}\frac{{\beta^{\alpha } }}{\Gamma \alpha }\frac{{\left( \gamma \right)^{\mu + l - 1} }}{{\overline{\gamma }^{\mu + l} }}e^{{\frac{{ - \mu \left( {1 + k} \right)\gamma }}{{\overline{\gamma }}}}} \left( \mu \right)^{\mu + 2l} \left( {\left( {1 + k} \right)} \right)^{\mu + l} \left( k \right)^{l} \mathop \smallint \limits_{0}^{\infty } t^{l - \alpha - 1} G^{{\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} }} \left( {\mu kt{|}\begin{array}{*{20}c} - \\ 0 \\ \end{array} } \right)G^{{\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} }} \left( {\frac{\beta }{t}{|}\begin{array}{*{20}c} - \\ 0 \\ \end{array} } \right)dt $$

(B.5)

Now applying the identity [25, Eq. (9.31.2)] and [28, Eq. (07.34.21.0011.01)] in Eq. (B.5). After a few simple algebraic manipulations we get PDF of the k-µ/inverse Gamma LOS shadowed fading model as in Eq. (6).

Appendix C: Establishing the Relationship Between the Inverse Gamma Shadowing Parameters (α, β) and (m, w)

The PDF of a Gamma random variable y parameterized in terms of (m_, b) [34, Eq. (2.55)], we get:

$$ f_{Y} \left( y \right) = \frac{1}{\Gamma m}(\frac{m}{{\text{b}}})^{m} y^{m - 1} e^{{ - \frac{my}{{\text{b}}}}} $$

(C.1)

where Γ(.) denotes the Gamma function, b is an average signal power and $m$ is an arbitrary shadowing parameter with values from 0.5 through infinity. The PDF of the inverse Gamma random variable $\uppsi$, parameterized in terms of parameters (m_, w) is obtained from PDF of the Gamma random variable y, parameterized in terms of parameters (m_, b), Eq. (C.1). Therefore, introducing a change of variables $\uppsi$ = 1/y and (w = b⁻¹) in the expression Eq. (C.1), we get:

$${f}_{\uppsi }\left(\uppsi \right)=\left|{f}_{Y}\left(y\right)\frac{dy}{d\uppsi }\right|\mathrm{at }(\mathrm{y }=\frac{1}{\uppsi })$$

(C.2)

Substituting Eq. (C.1) in Eq. (C.2),

$$ f_{{\uppsi }} \left( {\uppsi } \right) = \left| {\frac{1}{\Gamma m}(\frac{m}{b})^{m} y^{m - 1} e^{{ - \frac{my}{b}}} \left( { - \frac{1}{{{\uppsi }^{2} }}} \right)} \right|{\text{at }}\left( {{\text{y }} = \frac{1}{{\uppsi }}} \right) $$

(C.3)

Using the relation (w = b⁻¹) in Eq. (C.3), it can be written as

$$ f_{{\uppsi }} \left( {\uppsi } \right) = \frac{1}{\Gamma m}\left( {mw} \right)^{m} \left( {\frac{1}{{\uppsi }}} \right)^{m - 1} e^{{ - mw)\frac{1}{{\uppsi }}}} \left( {\frac{1}{{{\uppsi }^{2} }}} \right) $$

(C.4)

After arranging the terms in Eq. (C.4), we get Eq. (C.5)

$$ f_{{\uppsi }} \left( {\uppsi } \right) = \frac{{\left( {mw} \right)^{{m_{s} }} }}{\Gamma m}\frac{1}{{{\uppsi }^{m + 1} }}e^{{ - \frac{mw)}{{\uppsi }}}} $$

(C.5)

Now if we compare Eq. (C.5) with Eq. (B.2), the PDF of the inverse Gamma random variable $\uppsi$, ${f}_{\uppsi }\left(\uppsi \right)$ parameterized in terms of (α, β), then therelationship between the inverse Gamma shadowing parameters (α, β) and (m, w), comes out as (shape parameter α = m and scale parameter β = mw).